The $n$-fold reduced bar construction

Abstract

This paper is about a correspondence between monoidal structures in categories and $n$-fold loop spaces. We develop a new syntactical technique whose role is to substitute the coherence results, which were the main ingredients in the proofs that the Segal-Thomason bar construction provides an appropriate simplicial space. The results we present here enable more common categories to enter this delooping machine. For example, such is the category of finite sets with two monoidal structures brought by the disjoint union and Cartesian product.